LMFDB - Embedded newform 3006.2.a.s.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3006,2,Mod(1,3006)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3006, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3006 = 2 \cdot 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3006.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$24.0030308476$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.2777.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + x + 2 $ x^4 - x^3 - 4*x^2 + x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1002)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.679643$ of defining polynomial
Character	$\chi$	$=$	3006.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -0.416566 q^{5} +4.65960 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -0.416566 q^{5} +4.65960 q^{7} -1.00000 q^{8} +0.416566 q^{10} +0.283116 q^{13} -4.65960 q^{14} +1.00000 q^{16} -0.640714 q^{17} -5.88544 q^{19} -0.416566 q^{20} -5.88544 q^{23} -4.82647 q^{25} -0.283116 q^{26} +4.65960 q^{28} -5.16687 q^{29} +1.22584 q^{31} -1.00000 q^{32} +0.640714 q^{34} -1.94103 q^{35} -6.82816 q^{37} +5.88544 q^{38} +0.416566 q^{40} -12.4116 q^{41} -3.07617 q^{43} +5.88544 q^{46} -7.82647 q^{47} +14.7119 q^{49} +4.82647 q^{50} +0.283116 q^{52} +5.46888 q^{53} -4.65960 q^{56} +5.16687 q^{58} -3.20962 q^{59} -1.16687 q^{61} -1.22584 q^{62} +1.00000 q^{64} -0.117936 q^{65} -2.29863 q^{67} -0.640714 q^{68} +1.94103 q^{70} +8.60401 q^{71} +2.26690 q^{73} +6.82816 q^{74} -5.88544 q^{76} -7.69302 q^{79} -0.416566 q^{80} +12.4116 q^{82} -9.27646 q^{83} +0.266899 q^{85} +3.07617 q^{86} -3.94103 q^{89} +1.31921 q^{91} -5.88544 q^{92} +7.82647 q^{94} +2.45167 q^{95} +8.99334 q^{97} -14.7119 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} - 5 q^{5} + q^{7} - 4 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^4 - 5 * q^5 + q^7 - 4 * q^8	$ 4 q - 4 q^{2} + 4 q^{4} - 5 q^{5} + q^{7} - 4 q^{8} + 5 q^{10} + 8 q^{13} - q^{14} + 4 q^{16} - 10 q^{17} - 2 q^{19} - 5 q^{20} - 2 q^{23} + 5 q^{25} - 8 q^{26} + q^{28} - 14 q^{29} + q^{31} - 4 q^{32} + 10 q^{34} - 5 q^{35} + 5 q^{37} + 2 q^{38} + 5 q^{40} - 14 q^{41} + 2 q^{43} + 2 q^{46} - 7 q^{47} + 13 q^{49} - 5 q^{50} + 8 q^{52} - 3 q^{53} - q^{56} + 14 q^{58} + 5 q^{59} + 2 q^{61} - q^{62} + 4 q^{64} - 6 q^{65} - 7 q^{67} - 10 q^{68} + 5 q^{70} - 2 q^{71} + 2 q^{73} - 5 q^{74} - 2 q^{76} - 10 q^{79} - 5 q^{80} + 14 q^{82} - 13 q^{83} - 6 q^{85} - 2 q^{86} - 13 q^{89} - 30 q^{91} - 2 q^{92} + 7 q^{94} + 2 q^{95} + 5 q^{97} - 13 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 - 5 * q^5 + q^7 - 4 * q^8 + 5 * q^10 + 8 * q^13 - q^14 + 4 * q^16 - 10 * q^17 - 2 * q^19 - 5 * q^20 - 2 * q^23 + 5 * q^25 - 8 * q^26 + q^28 - 14 * q^29 + q^31 - 4 * q^32 + 10 * q^34 - 5 * q^35 + 5 * q^37 + 2 * q^38 + 5 * q^40 - 14 * q^41 + 2 * q^43 + 2 * q^46 - 7 * q^47 + 13 * q^49 - 5 * q^50 + 8 * q^52 - 3 * q^53 - q^56 + 14 * q^58 + 5 * q^59 + 2 * q^61 - q^62 + 4 * q^64 - 6 * q^65 - 7 * q^67 - 10 * q^68 + 5 * q^70 - 2 * q^71 + 2 * q^73 - 5 * q^74 - 2 * q^76 - 10 * q^79 - 5 * q^80 + 14 * q^82 - 13 * q^83 - 6 * q^85 - 2 * q^86 - 13 * q^89 - 30 * q^91 - 2 * q^92 + 7 * q^94 + 2 * q^95 + 5 * q^97 - 13 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−0.416566	−0.186294	−0.0931469	−	0.995652i	$-0.529693\pi$
$5$	−0.416566	−0.186294	−0.0931469	+	0.995652i	$0.529693\pi$
$6$	0	0
$7$	4.65960	1.76116	0.880582	−	0.473893i	$-0.157152\pi$
$7$	4.65960	1.76116	0.880582	+	0.473893i	$0.157152\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0.416566	0.131730
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0.283116	0.0785223	0.0392611	−	0.999229i	$-0.487500\pi$
$13$	0.283116	0.0785223	0.0392611	+	0.999229i	$0.487500\pi$
$14$	−4.65960	−1.24533
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.640714	−0.155396	−0.0776979	−	0.996977i	$-0.524757\pi$
$17$	−0.640714	−0.155396	−0.0776979	+	0.996977i	$0.524757\pi$
$18$	0	0
$19$	−5.88544	−1.35021	−0.675106	−	0.737720i	$-0.735902\pi$
$19$	−5.88544	−1.35021	−0.675106	+	0.737720i	$0.735902\pi$
$20$	−0.416566	−0.0931469
$21$	0	0
$22$	0	0
$23$	−5.88544	−1.22720	−0.613600	−	0.789617i	$-0.710279\pi$
$23$	−5.88544	−1.22720	−0.613600	+	0.789617i	$0.710279\pi$
$24$	0	0
$25$	−4.82647	−0.965295
$26$	−0.283116	−0.0555236
$27$	0	0
$28$	4.65960	0.880582
$29$	−5.16687	−0.959463	−0.479732	−	0.877415i	$-0.659266\pi$
$29$	−5.16687	−0.959463	−0.479732	+	0.877415i	$0.659266\pi$
$30$	0	0
$31$	1.22584	0.220167	0.110083	−	0.993922i	$-0.464888\pi$
$31$	1.22584	0.220167	0.110083	+	0.993922i	$0.464888\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0.640714	0.109881
$35$	−1.94103	−0.328094
$36$	0	0
$37$	−6.82816	−1.12254	−0.561271	−	0.827632i	$-0.689688\pi$
$37$	−6.82816	−1.12254	−0.561271	+	0.827632i	$0.689688\pi$
$38$	5.88544	0.954745
$39$	0	0
$40$	0.416566	0.0658648
$41$	−12.4116	−1.93837	−0.969183	−	0.246343i	$-0.920771\pi$
$41$	−12.4116	−1.93837	−0.969183	+	0.246343i	$0.920771\pi$
$42$	0	0
$43$	−3.07617	−0.469112	−0.234556	−	0.972103i	$-0.575364\pi$
$43$	−3.07617	−0.469112	−0.234556	+	0.972103i	$0.575364\pi$
$44$	0	0
$45$	0	0
$46$	5.88544	0.867761
$47$	−7.82647	−1.14161	−0.570804	−	0.821086i	$-0.693368\pi$
$47$	−7.82647	−1.14161	−0.570804	+	0.821086i	$0.693368\pi$
$48$	0	0
$49$	14.7119	2.10170
$50$	4.82647	0.682566
$51$	0	0
$52$	0.283116	0.0392611
$53$	5.46888	0.751208	0.375604	−	0.926780i	$-0.377435\pi$
$53$	5.46888	0.751208	0.375604	+	0.926780i	$0.377435\pi$
$54$	0	0
$55$	0	0
$56$	−4.65960	−0.622666
$57$	0	0
$58$	5.16687	0.678443
$59$	−3.20962	−0.417857	−0.208928	−	0.977931i	$-0.566998\pi$
$59$	−3.20962	−0.417857	−0.208928	+	0.977931i	$0.566998\pi$
$60$	0	0
$61$	−1.16687	−0.149402	−0.0747011	−	0.997206i	$-0.523800\pi$
$61$	−1.16687	−0.149402	−0.0747011	+	0.997206i	$0.523800\pi$
$62$	−1.22584	−0.155681
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.117936	−0.0146282
$66$	0	0
$67$	−2.29863	−0.280822	−0.140411	−	0.990093i	$-0.544842\pi$
$67$	−2.29863	−0.280822	−0.140411	+	0.990093i	$0.544842\pi$
$68$	−0.640714	−0.0776979
$69$	0	0
$70$	1.94103	0.231998
$71$	8.60401	1.02111	0.510554	−	0.859846i	$-0.329440\pi$
$71$	8.60401	1.02111	0.510554	+	0.859846i	$0.329440\pi$
$72$	0	0
$73$	2.26690	0.265321	0.132660	−	0.991162i	$-0.457648\pi$
$73$	2.26690	0.265321	0.132660	+	0.991162i	$0.457648\pi$
$74$	6.82816	0.793758
$75$	0	0
$76$	−5.88544	−0.675106
$77$	0	0
$78$	0	0
$79$	−7.69302	−0.865533	−0.432766	−	0.901506i	$-0.642462\pi$
$79$	−7.69302	−0.865533	−0.432766	+	0.901506i	$0.642462\pi$
$80$	−0.416566	−0.0465735
$81$	0	0
$82$	12.4116	1.37063
$83$	−9.27646	−1.01822	−0.509112	−	0.860700i	$-0.670026\pi$
$83$	−9.27646	−1.01822	−0.509112	+	0.860700i	$0.670026\pi$
$84$	0	0
$85$	0.266899	0.0289493
$86$	3.07617	0.331712
$87$	0	0
$88$	0	0
$89$	−3.94103	−0.417749	−0.208874	−	0.977943i	$-0.566980\pi$
$89$	−3.94103	−0.417749	−0.208874	+	0.977943i	$0.566980\pi$
$90$	0	0
$91$	1.31921	0.138291
$92$	−5.88544	−0.613600
$93$	0	0
$94$	7.82647	0.807239
$95$	2.45167	0.251536
$96$	0	0
$97$	8.99334	0.913135	0.456568	−	0.889689i	$-0.349079\pi$
$97$	8.99334	0.913135	0.456568	+	0.889689i	$0.349079\pi$
$98$	−14.7119	−1.48613
$99$	0	0
$100$	−4.82647	−0.482647
$101$	12.3364	1.22752	0.613759	−	0.789493i	$-0.289656\pi$
$101$	12.3364	1.22752	0.613759	+	0.789493i	$0.289656\pi$
$102$	0	0
$103$	−12.3448	−1.21637	−0.608183	−	0.793797i	$-0.708101\pi$
$103$	−12.3448	−1.21637	−0.608183	+	0.793797i	$0.708101\pi$
$104$	−0.283116	−0.0277618
$105$	0	0
$106$	−5.46888	−0.531184
$107$	19.0867	1.84518	0.922591	−	0.385779i	$-0.126067\pi$
$107$	19.0867	1.84518	0.922591	+	0.385779i	$0.126067\pi$
$108$	0	0
$109$	17.2553	1.65276	0.826378	−	0.563116i	$-0.190398\pi$
$109$	17.2553	1.65276	0.826378	+	0.563116i	$0.190398\pi$
$110$	0	0
$111$	0	0
$112$	4.65960	0.440291
$113$	−2.91099	−0.273843	−0.136921	−	0.990582i	$-0.543721\pi$
$113$	−2.91099	−0.273843	−0.136921	+	0.990582i	$0.543721\pi$
$114$	0	0
$115$	2.45167	0.228620
$116$	−5.16687	−0.479732
$117$	0	0
$118$	3.20962	0.295469
$119$	−2.98547	−0.273678
$120$	0	0
$121$	−11.0000	−1.00000
$122$	1.16687	0.105643
$123$	0	0
$124$	1.22584	0.110083
$125$	4.09337	0.366122
$126$	0	0
$127$	−21.2636	−1.88684	−0.943421	−	0.331599i	$-0.892412\pi$
$127$	−21.2636	−1.88684	−0.943421	+	0.331599i	$0.892412\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0.117936	0.0103437
$131$	14.2030	1.24092	0.620459	−	0.784239i	$-0.286946\pi$
$131$	14.2030	1.24092	0.620459	+	0.784239i	$0.286946\pi$
$132$	0	0
$133$	−27.4238	−2.37795
$134$	2.29863	0.198571
$135$	0	0
$136$	0.640714	0.0549407
$137$	10.5484	0.901213	0.450606	−	0.892723i	$-0.351208\pi$
$137$	10.5484	0.901213	0.450606	+	0.892723i	$0.351208\pi$
$138$	0	0
$139$	11.9550	1.01401	0.507003	−	0.861944i	$-0.330753\pi$
$139$	11.9550	1.01401	0.507003	+	0.861944i	$0.330753\pi$
$140$	−1.94103	−0.164047
$141$	0	0
$142$	−8.60401	−0.722033
$143$	0	0
$144$	0	0
$145$	2.15234	0.178742
$146$	−2.26690	−0.187610
$147$	0	0
$148$	−6.82816	−0.561271
$149$	−3.58343	−0.293566	−0.146783	−	0.989169i	$-0.546892\pi$
$149$	−3.58343	−0.293566	−0.146783	+	0.989169i	$0.546892\pi$
$150$	0	0
$151$	−3.74074	−0.304417	−0.152209	−	0.988348i	$-0.548639\pi$
$151$	−3.74074	−0.304417	−0.152209	+	0.988348i	$0.548639\pi$
$152$	5.88544	0.477372
$153$	0	0
$154$	0	0
$155$	−0.510642	−0.0410157
$156$	0	0
$157$	−13.7709	−1.09904	−0.549518	−	0.835482i	$-0.685189\pi$
$157$	−13.7709	−1.09904	−0.549518	+	0.835482i	$0.685189\pi$
$158$	7.69302	0.612024
$159$	0	0
$160$	0.416566	0.0329324
$161$	−27.4238	−2.16130
$162$	0	0
$163$	11.2398	0.880366	0.440183	−	0.897908i	$-0.354914\pi$
$163$	11.2398	0.880366	0.440183	+	0.897908i	$0.354914\pi$
$164$	−12.4116	−0.969183
$165$	0	0
$166$	9.27646	0.719993
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−12.9198	−0.993834
$170$	−0.266899	−0.0204702
$171$	0	0
$172$	−3.07617	−0.234556
$173$	−1.16687	−0.0887154	−0.0443577	−	0.999016i	$-0.514124\pi$
$173$	−1.16687	−0.0887154	−0.0443577	+	0.999016i	$0.514124\pi$
$174$	0	0
$175$	−22.4895	−1.70004
$176$	0	0
$177$	0	0
$178$	3.94103	0.295393
$179$	−7.77088	−0.580823	−0.290412	−	0.956902i	$-0.593792\pi$
$179$	−7.77088	−0.580823	−0.290412	+	0.956902i	$0.593792\pi$
$180$	0	0
$181$	10.2570	0.762394	0.381197	−	0.924494i	$-0.375512\pi$
$181$	10.2570	0.762394	0.381197	+	0.924494i	$0.375512\pi$
$182$	−1.31921	−0.0977863
$183$	0	0
$184$	5.88544	0.433880
$185$	2.84438	0.209123
$186$	0	0
$187$	0	0
$188$	−7.82647	−0.570804
$189$	0	0
$190$	−2.45167	−0.177863
$191$	13.5630	0.981381	0.490690	−	0.871334i	$-0.336745\pi$
$191$	13.5630	0.981381	0.490690	+	0.871334i	$0.336745\pi$
$192$	0	0
$193$	−2.56623	−0.184721	−0.0923607	−	0.995726i	$-0.529441\pi$
$193$	−2.56623	−0.184721	−0.0923607	+	0.995726i	$0.529441\pi$
$194$	−8.99334	−0.645684
$195$	0	0
$196$	14.7119	1.05085
$197$	2.92383	0.208314	0.104157	−	0.994561i	$-0.466786\pi$
$197$	2.92383	0.208314	0.104157	+	0.994561i	$0.466786\pi$
$198$	0	0
$199$	−3.43377	−0.243413	−0.121707	−	0.992566i	$-0.538837\pi$
$199$	−3.43377	−0.243413	−0.121707	+	0.992566i	$0.538837\pi$
$200$	4.82647	0.341283
$201$	0	0
$202$	−12.3364	−0.867987
$203$	−24.0756	−1.68977
$204$	0	0
$205$	5.17025	0.361106
$206$	12.3448	0.860100
$207$	0	0
$208$	0.283116	0.0196306
$209$	0	0
$210$	0	0
$211$	−21.8854	−1.50666	−0.753328	−	0.657645i	$-0.771553\pi$
$211$	−21.8854	−1.50666	−0.753328	+	0.657645i	$0.771553\pi$
$212$	5.46888	0.375604
$213$	0	0
$214$	−19.0867	−1.30474
$215$	1.28143	0.0873926
$216$	0	0
$217$	5.71191	0.387750
$218$	−17.2553	−1.16867
$219$	0	0
$220$	0	0
$221$	−0.181396	−0.0122020
$222$	0	0
$223$	−4.65960	−0.312030	−0.156015	−	0.987755i	$-0.549865\pi$
$223$	−4.65960	−0.312030	−0.156015	+	0.987755i	$0.549865\pi$
$224$	−4.65960	−0.311333
$225$	0	0
$226$	2.91099	0.193636
$227$	22.4467	1.48984	0.744920	−	0.667154i	$-0.232488\pi$
$227$	22.4467	1.48984	0.744920	+	0.667154i	$0.232488\pi$
$228$	0	0
$229$	12.3715	0.817533	0.408766	−	0.912639i	$-0.365959\pi$
$229$	12.3715	0.817533	0.408766	+	0.912639i	$0.365959\pi$
$230$	−2.45167	−0.161659
$231$	0	0
$232$	5.16687	0.339222
$233$	−19.5940	−1.28364	−0.641822	−	0.766854i	$-0.721821\pi$
$233$	−19.5940	−1.28364	−0.641822	+	0.766854i	$0.721821\pi$
$234$	0	0
$235$	3.26024	0.212675
$236$	−3.20962	−0.208928
$237$	0	0
$238$	2.98547	0.193519
$239$	−3.43377	−0.222112	−0.111056	−	0.993814i	$-0.535423\pi$
$239$	−3.43377	−0.222112	−0.111056	+	0.993814i	$0.535423\pi$
$240$	0	0
$241$	14.4861	0.933130	0.466565	−	0.884487i	$-0.345491\pi$
$241$	14.4861	0.933130	0.466565	+	0.884487i	$0.345491\pi$
$242$	11.0000	0.707107
$243$	0	0
$244$	−1.16687	−0.0747011
$245$	−6.12848	−0.391534
$246$	0	0
$247$	−1.66626	−0.106022
$248$	−1.22584	−0.0778407
$249$	0	0
$250$	−4.09337	−0.258888
$251$	26.8576	1.69524	0.847618	−	0.530607i	$-0.178036\pi$
$251$	26.8576	1.69524	0.847618	+	0.530607i	$0.178036\pi$
$252$	0	0
$253$	0	0
$254$	21.2636	1.33420
$255$	0	0
$256$	1.00000	0.0625000
$257$	3.57846	0.223218	0.111609	−	0.993752i	$-0.464400\pi$
$257$	3.57846	0.223218	0.111609	+	0.993752i	$0.464400\pi$
$258$	0	0
$259$	−31.8165	−1.97698
$260$	−0.117936	−0.00731411
$261$	0	0
$262$	−14.2030	−0.877462
$263$	−5.52517	−0.340697	−0.170348	−	0.985384i	$-0.554489\pi$
$263$	−5.52517	−0.340697	−0.170348	+	0.985384i	$0.554489\pi$
$264$	0	0
$265$	−2.27815	−0.139945
$266$	27.4238	1.68146
$267$	0	0
$268$	−2.29863	−0.140411
$269$	−29.0039	−1.76840	−0.884199	−	0.467110i	$-0.845295\pi$
$269$	−29.0039	−1.76840	−0.884199	+	0.467110i	$0.845295\pi$
$270$	0	0
$271$	−24.0301	−1.45973	−0.729863	−	0.683593i	$-0.760416\pi$
$271$	−24.0301	−1.45973	−0.729863	+	0.683593i	$0.760416\pi$
$272$	−0.640714	−0.0388490
$273$	0	0
$274$	−10.5484	−0.637254
$275$	0	0
$276$	0	0
$277$	2.99382	0.179881	0.0899406	−	0.995947i	$-0.471332\pi$
$277$	2.99382	0.179881	0.0899406	+	0.995947i	$0.471332\pi$
$278$	−11.9550	−0.717010
$279$	0	0
$280$	1.94103	0.115999
$281$	−1.72523	−0.102919	−0.0514593	−	0.998675i	$-0.516387\pi$
$281$	−1.72523	−0.102919	−0.0514593	+	0.998675i	$0.516387\pi$
$282$	0	0
$283$	18.5372	1.10192	0.550960	−	0.834531i	$-0.314262\pi$
$283$	18.5372	1.10192	0.550960	+	0.834531i	$0.314262\pi$
$284$	8.60401	0.510554
$285$	0	0
$286$	0	0
$287$	−57.8331	−3.41378
$288$	0	0
$289$	−16.5895	−0.975852
$290$	−2.15234	−0.126390
$291$	0	0
$292$	2.26690	0.132660
$293$	−14.8331	−0.866561	−0.433280	−	0.901259i	$-0.642644\pi$
$293$	−14.8331	−0.866561	−0.433280	+	0.901259i	$0.642644\pi$
$294$	0	0
$295$	1.33702	0.0778442
$296$	6.82816	0.396879
$297$	0	0
$298$	3.58343	0.207583
$299$	−1.66626	−0.0963625
$300$	0	0
$301$	−14.3337	−0.826183
$302$	3.74074	0.215256
$303$	0	0
$304$	−5.88544	−0.337553
$305$	0.486077	0.0278327
$306$	0	0
$307$	−31.9927	−1.82592	−0.912961	−	0.408047i	$-0.866210\pi$
$307$	−31.9927	−1.82592	−0.912961	+	0.408047i	$0.866210\pi$
$308$	0	0
$309$	0	0
$310$	0.510642	0.0290025
$311$	7.07884	0.401404	0.200702	−	0.979652i	$-0.435678\pi$
$311$	7.07884	0.401404	0.200702	+	0.979652i	$0.435678\pi$
$312$	0	0
$313$	23.6496	1.33675	0.668376	−	0.743823i	$-0.266990\pi$
$313$	23.6496	1.33675	0.668376	+	0.743823i	$0.266990\pi$
$314$	13.7709	0.777136
$315$	0	0
$316$	−7.69302	−0.432766
$317$	13.7007	0.769506	0.384753	−	0.923020i	$-0.374287\pi$
$317$	13.7007	0.769506	0.384753	+	0.923020i	$0.374287\pi$
$318$	0	0
$319$	0	0
$320$	−0.416566	−0.0232867
$321$	0	0
$322$	27.4238	1.52827
$323$	3.77088	0.209818
$324$	0	0
$325$	−1.36645	−0.0757971
$326$	−11.2398	−0.622513
$327$	0	0
$328$	12.4116	0.685316
$329$	−36.4683	−2.01056
$330$	0	0
$331$	−9.40991	−0.517215	−0.258608	−	0.965982i	$-0.583264\pi$
$331$	−9.40991	−0.517215	−0.258608	+	0.965982i	$0.583264\pi$
$332$	−9.27646	−0.509112
$333$	0	0
$334$	−1.00000	−0.0547176
$335$	0.957530	0.0523155
$336$	0	0
$337$	21.9345	1.19485	0.597423	−	0.801926i	$-0.296191\pi$
$337$	21.9345	1.19485	0.597423	+	0.801926i	$0.296191\pi$
$338$	12.9198	0.702747
$339$	0	0
$340$	0.266899	0.0144747
$341$	0	0
$342$	0	0
$343$	35.9345	1.94028
$344$	3.07617	0.165856
$345$	0	0
$346$	1.16687	0.0627312
$347$	−16.6468	−0.893645	−0.446823	−	0.894623i	$-0.647444\pi$
$347$	−16.6468	−0.893645	−0.446823	+	0.894623i	$0.647444\pi$
$348$	0	0
$349$	4.49442	0.240581	0.120291	−	0.992739i	$-0.461617\pi$
$349$	4.49442	0.240581	0.120291	+	0.992739i	$0.461617\pi$
$350$	22.4895	1.20211
$351$	0	0
$352$	0	0
$353$	−9.72523	−0.517622	−0.258811	−	0.965928i	$-0.583331\pi$
$353$	−9.72523	−0.517622	−0.258811	+	0.965928i	$0.583331\pi$
$354$	0	0
$355$	−3.58414	−0.190226
$356$	−3.94103	−0.208874
$357$	0	0
$358$	7.77088	0.410704
$359$	9.61854	0.507647	0.253824	−	0.967251i	$-0.418312\pi$
$359$	9.61854	0.507647	0.253824	+	0.967251i	$0.418312\pi$
$360$	0	0
$361$	15.6384	0.823075
$362$	−10.2570	−0.539094
$363$	0	0
$364$	1.31921	0.0691453
$365$	−0.944313	−0.0494276
$366$	0	0
$367$	14.1523	0.738746	0.369373	−	0.929281i	$-0.379572\pi$
$367$	14.1523	0.738746	0.369373	+	0.929281i	$0.379572\pi$
$368$	−5.88544	−0.306800
$369$	0	0
$370$	−2.84438	−0.147872
$371$	25.4828	1.32300
$372$	0	0
$373$	−28.1149	−1.45574	−0.727868	−	0.685717i	$-0.759489\pi$
$373$	−28.1149	−1.45574	−0.727868	+	0.685717i	$0.759489\pi$
$374$	0	0
$375$	0	0
$376$	7.82647	0.403620
$377$	−1.46282	−0.0753392
$378$	0	0
$379$	16.5212	0.848636	0.424318	−	0.905513i	$-0.360514\pi$
$379$	16.5212	0.848636	0.424318	+	0.905513i	$0.360514\pi$
$380$	2.45167	0.125768
$381$	0	0
$382$	−13.5630	−0.693941
$383$	15.6907	0.801759	0.400879	−	0.916131i	$-0.368705\pi$
$383$	15.6907	0.801759	0.400879	+	0.916131i	$0.368705\pi$
$384$	0	0
$385$	0	0
$386$	2.56623	0.130618
$387$	0	0
$388$	8.99334	0.456568
$389$	21.1842	1.07408	0.537040	−	0.843557i	$-0.319542\pi$
$389$	21.1842	1.07408	0.537040	+	0.843557i	$0.319542\pi$
$390$	0	0
$391$	3.77088	0.190702
$392$	−14.7119	−0.743064
$393$	0	0
$394$	−2.92383	−0.147300
$395$	3.20465	0.161243
$396$	0	0
$397$	−2.03778	−0.102273	−0.0511367	−	0.998692i	$-0.516284\pi$
$397$	−2.03778	−0.102273	−0.0511367	+	0.998692i	$0.516284\pi$
$398$	3.43377	0.172119
$399$	0	0
$400$	−4.82647	−0.241324
$401$	11.3971	0.569142	0.284571	−	0.958655i	$-0.408149\pi$
$401$	11.3971	0.569142	0.284571	+	0.958655i	$0.408149\pi$
$402$	0	0
$403$	0.347054	0.0172880
$404$	12.3364	0.613759
$405$	0	0
$406$	24.0756	1.19485
$407$	0	0
$408$	0	0
$409$	−18.1012	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$409$	−18.1012	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$410$	−5.17025	−0.255340
$411$	0	0
$412$	−12.3448	−0.608183
$413$	−14.9556	−0.735915
$414$	0	0
$415$	3.86425	0.189689
$416$	−0.283116	−0.0138809
$417$	0	0
$418$	0	0
$419$	−15.2047	−0.742796	−0.371398	−	0.928474i	$-0.621121\pi$
$419$	−15.2047	−0.742796	−0.371398	+	0.928474i	$0.621121\pi$
$420$	0	0
$421$	16.2914	0.793992	0.396996	−	0.917820i	$-0.370053\pi$
$421$	16.2914	0.793992	0.396996	+	0.917820i	$0.370053\pi$
$422$	21.8854	1.06537
$423$	0	0
$424$	−5.46888	−0.265592
$425$	3.09239	0.150003
$426$	0	0
$427$	−5.43715	−0.263122
$428$	19.0867	0.922591
$429$	0	0
$430$	−1.28143	−0.0617959
$431$	−9.53052	−0.459069	−0.229534	−	0.973301i	$-0.573720\pi$
$431$	−9.53052	−0.459069	−0.229534	+	0.973301i	$0.573720\pi$
$432$	0	0
$433$	−28.5027	−1.36975	−0.684876	−	0.728660i	$-0.740143\pi$
$433$	−28.5027	−1.36975	−0.684876	+	0.728660i	$0.740143\pi$
$434$	−5.71191	−0.274181
$435$	0	0
$436$	17.2553	0.826378
$437$	34.6384	1.65698
$438$	0	0
$439$	−38.0222	−1.81470	−0.907350	−	0.420377i	$-0.861898\pi$
$439$	−38.0222	−1.81470	−0.907350	+	0.420377i	$0.861898\pi$
$440$	0	0
$441$	0	0
$442$	0.181396	0.00862814
$443$	−6.29633	−0.299148	−0.149574	−	0.988751i	$-0.547790\pi$
$443$	−6.29633	−0.299148	−0.149574	+	0.988751i	$0.547790\pi$
$444$	0	0
$445$	1.64170	0.0778240
$446$	4.65960	0.220639
$447$	0	0
$448$	4.65960	0.220146
$449$	−20.0967	−0.948424	−0.474212	−	0.880411i	$-0.657267\pi$
$449$	−20.0967	−0.948424	−0.474212	+	0.880411i	$0.657267\pi$
$450$	0	0
$451$	0	0
$452$	−2.91099	−0.136921
$453$	0	0
$454$	−22.4467	−1.05348
$455$	−0.549537	−0.0257627
$456$	0	0
$457$	30.0278	1.40464	0.702322	−	0.711860i	$-0.252147\pi$
$457$	30.0278	1.40464	0.702322	+	0.711860i	$0.252147\pi$
$458$	−12.3715	−0.578083
$459$	0	0
$460$	2.45167	0.114310
$461$	1.21459	0.0565691	0.0282845	−	0.999600i	$-0.490996\pi$
$461$	1.21459	0.0565691	0.0282845	+	0.999600i	$0.490996\pi$
$462$	0	0
$463$	6.80637	0.316319	0.158159	−	0.987414i	$-0.449444\pi$
$463$	6.80637	0.316319	0.158159	+	0.987414i	$0.449444\pi$
$464$	−5.16687	−0.239866
$465$	0	0
$466$	19.5940	0.907673
$467$	−16.5563	−0.766134	−0.383067	−	0.923721i	$-0.625132\pi$
$467$	−16.5563	−0.766134	−0.383067	+	0.923721i	$0.625132\pi$
$468$	0	0
$469$	−10.7107	−0.494574
$470$	−3.26024	−0.150384
$471$	0	0
$472$	3.20962	0.147735
$473$	0	0
$474$	0	0
$475$	28.4059	1.30335
$476$	−2.98547	−0.136839
$477$	0	0
$478$	3.43377	0.157057
$479$	−36.6629	−1.67517	−0.837585	−	0.546307i	$-0.816033\pi$
$479$	−36.6629	−1.67517	−0.837585	+	0.546307i	$0.816033\pi$
$480$	0	0
$481$	−1.93316	−0.0881446
$482$	−14.4861	−0.659823
$483$	0	0
$484$	−11.0000	−0.500000
$485$	−3.74632	−0.170112
$486$	0	0
$487$	9.01223	0.408383	0.204192	−	0.978931i	$-0.434543\pi$
$487$	9.01223	0.408383	0.204192	+	0.978931i	$0.434543\pi$
$488$	1.16687	0.0528217
$489$	0	0
$490$	6.12848	0.276857
$491$	34.5430	1.55890	0.779451	−	0.626463i	$-0.215498\pi$
$491$	34.5430	1.55890	0.779451	+	0.626463i	$0.215498\pi$
$492$	0	0
$493$	3.31048	0.149097
$494$	1.66626	0.0749687
$495$	0	0
$496$	1.22584	0.0550417
$497$	40.0913	1.79834
$498$	0	0
$499$	−35.3331	−1.58173	−0.790864	−	0.611992i	$-0.790369\pi$
$499$	−35.3331	−1.58173	−0.790864	+	0.611992i	$0.790369\pi$
$500$	4.09337	0.183061
$501$	0	0
$502$	−26.8576	−1.19871
$503$	14.0245	0.625320	0.312660	−	0.949865i	$-0.398780\pi$
$503$	14.0245	0.625320	0.312660	+	0.949865i	$0.398780\pi$
$504$	0	0
$505$	−5.13893	−0.228679
$506$	0	0
$507$	0	0
$508$	−21.2636	−0.943421
$509$	7.01453	0.310913	0.155457	−	0.987843i	$-0.450315\pi$
$509$	7.01453	0.310913	0.155457	+	0.987843i	$0.450315\pi$
$510$	0	0
$511$	10.5629	0.467273
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−3.57846	−0.157839
$515$	5.14240	0.226601
$516$	0	0
$517$	0	0
$518$	31.8165	1.39794
$519$	0	0
$520$	0.117936	0.00517186
$521$	−3.70634	−0.162378	−0.0811889	−	0.996699i	$-0.525872\pi$
$521$	−3.70634	−0.162378	−0.0811889	+	0.996699i	$0.525872\pi$
$522$	0	0
$523$	34.4417	1.50603	0.753016	−	0.658002i	$-0.228598\pi$
$523$	34.4417	1.50603	0.753016	+	0.658002i	$0.228598\pi$
$524$	14.2030	0.620459
$525$	0	0
$526$	5.52517	0.240909
$527$	−0.785410	−0.0342130
$528$	0	0
$529$	11.6384	0.506018
$530$	2.27815	0.0989564
$531$	0	0
$532$	−27.4238	−1.18897
$533$	−3.51392	−0.152205
$534$	0	0
$535$	−7.95087	−0.343746
$536$	2.29863	0.0992857
$537$	0	0
$538$	29.0039	1.25045
$539$	0	0
$540$	0	0
$541$	2.07378	0.0891587	0.0445793	−	0.999006i	$-0.485805\pi$
$541$	2.07378	0.0891587	0.0445793	+	0.999006i	$0.485805\pi$
$542$	24.0301	1.03218
$543$	0	0
$544$	0.640714	0.0274704
$545$	−7.18796	−0.307898
$546$	0	0
$547$	37.0928	1.58597	0.792986	−	0.609240i	$-0.208525\pi$
$547$	37.0928	1.58597	0.792986	+	0.609240i	$0.208525\pi$
$548$	10.5484	0.450606
$549$	0	0
$550$	0	0
$551$	30.4093	1.29548
$552$	0	0
$553$	−35.8464	−1.52435
$554$	−2.99382	−0.127195
$555$	0	0
$556$	11.9550	0.507003
$557$	−8.75635	−0.371019	−0.185509	−	0.982643i	$-0.559393\pi$
$557$	−8.75635	−0.371019	−0.185509	+	0.982643i	$0.559393\pi$
$558$	0	0
$559$	−0.870913	−0.0368357
$560$	−1.94103	−0.0820236
$561$	0	0
$562$	1.72523	0.0727745
$563$	2.41924	0.101959	0.0509794	−	0.998700i	$-0.483766\pi$
$563$	2.41924	0.101959	0.0509794	+	0.998700i	$0.483766\pi$
$564$	0	0
$565$	1.21262	0.0510153
$566$	−18.5372	−0.779176
$567$	0	0
$568$	−8.60401	−0.361016
$569$	1.68627	0.0706920	0.0353460	−	0.999375i	$-0.488747\pi$
$569$	1.68627	0.0706920	0.0353460	+	0.999375i	$0.488747\pi$
$570$	0	0
$571$	−1.66359	−0.0696190	−0.0348095	−	0.999394i	$-0.511082\pi$
$571$	−1.66359	−0.0696190	−0.0348095	+	0.999394i	$0.511082\pi$
$572$	0	0
$573$	0	0
$574$	57.8331	2.41391
$575$	28.4059	1.18461
$576$	0	0
$577$	−31.7020	−1.31977	−0.659885	−	0.751366i	$-0.729395\pi$
$577$	−31.7020	−1.31977	−0.659885	+	0.751366i	$0.729395\pi$
$578$	16.5895	0.690032
$579$	0	0
$580$	2.15234	0.0893711
$581$	−43.2246	−1.79326
$582$	0	0
$583$	0	0
$584$	−2.26690	−0.0938050
$585$	0	0
$586$	14.8331	0.612751
$587$	−23.0817	−0.952686	−0.476343	−	0.879260i	$-0.658038\pi$
$587$	−23.0817	−0.952686	−0.476343	+	0.879260i	$0.658038\pi$
$588$	0	0
$589$	−7.21459	−0.297272
$590$	−1.33702	−0.0550442
$591$	0	0
$592$	−6.82816	−0.280636
$593$	−21.0156	−0.863008	−0.431504	−	0.902111i	$-0.642017\pi$
$593$	−21.0156	−0.863008	−0.431504	+	0.902111i	$0.642017\pi$
$594$	0	0
$595$	1.24365	0.0509845
$596$	−3.58343	−0.146783
$597$	0	0
$598$	1.66626	0.0681386
$599$	0.274769	0.0112267	0.00561337	−	0.999984i	$-0.498213\pi$
$599$	0.274769	0.0112267	0.00561337	+	0.999984i	$0.498213\pi$
$600$	0	0
$601$	−42.1590	−1.71970	−0.859851	−	0.510546i	$-0.829443\pi$
$601$	−42.1590	−1.71970	−0.859851	+	0.510546i	$0.829443\pi$
$602$	14.3337	0.584200
$603$	0	0
$604$	−3.74074	−0.152209
$605$	4.58222	0.186294
$606$	0	0
$607$	36.3658	1.47604	0.738022	−	0.674777i	$-0.235760\pi$
$607$	36.3658	1.47604	0.738022	+	0.674777i	$0.235760\pi$
$608$	5.88544	0.238686
$609$	0	0
$610$	−0.486077	−0.0196807
$611$	−2.21580	−0.0896417
$612$	0	0
$613$	−28.3291	−1.14420	−0.572102	−	0.820183i	$-0.693872\pi$
$613$	−28.3291	−1.14420	−0.572102	+	0.820183i	$0.693872\pi$
$614$	31.9927	1.29112
$615$	0	0
$616$	0	0
$617$	−44.7941	−1.80334	−0.901672	−	0.432421i	$-0.857660\pi$
$617$	−44.7941	−1.80334	−0.901672	+	0.432421i	$0.857660\pi$
$618$	0	0
$619$	43.9682	1.76723	0.883615	−	0.468214i	$-0.155102\pi$
$619$	43.9682	1.76723	0.883615	+	0.468214i	$0.155102\pi$
$620$	−0.510642	−0.0205079
$621$	0	0
$622$	−7.07884	−0.283836
$623$	−18.3636	−0.735724
$624$	0	0
$625$	22.4272	0.897088
$626$	−23.6496	−0.945227
$627$	0	0
$628$	−13.7709	−0.549518
$629$	4.37490	0.174439
$630$	0	0
$631$	15.8265	0.630042	0.315021	−	0.949085i	$-0.397988\pi$
$631$	15.8265	0.630042	0.315021	+	0.949085i	$0.397988\pi$
$632$	7.69302	0.306012
$633$	0	0
$634$	−13.7007	−0.544123
$635$	8.85770	0.351507
$636$	0	0
$637$	4.16518	0.165030
$638$	0	0
$639$	0	0
$640$	0.416566	0.0164662
$641$	25.4792	1.00637	0.503184	−	0.864179i	$-0.332162\pi$
$641$	25.4792	1.00637	0.503184	+	0.864179i	$0.332162\pi$
$642$	0	0
$643$	20.6623	0.814841	0.407420	−	0.913241i	$-0.366428\pi$
$643$	20.6623	0.814841	0.407420	+	0.913241i	$0.366428\pi$
$644$	−27.4238	−1.08065
$645$	0	0
$646$	−3.77088	−0.148363
$647$	−1.24899	−0.0491030	−0.0245515	−	0.999699i	$-0.507816\pi$
$647$	−1.24899	−0.0491030	−0.0245515	+	0.999699i	$0.507816\pi$
$648$	0	0
$649$	0	0
$650$	1.36645	0.0535967
$651$	0	0
$652$	11.2398	0.440183
$653$	−3.27149	−0.128023	−0.0640116	−	0.997949i	$-0.520389\pi$
$653$	−3.27149	−0.128023	−0.0640116	+	0.997949i	$0.520389\pi$
$654$	0	0
$655$	−5.91647	−0.231176
$656$	−12.4116	−0.484591
$657$	0	0
$658$	36.4683	1.42168
$659$	20.0805	0.782227	0.391113	−	0.920343i	$-0.372090\pi$
$659$	20.0805	0.782227	0.391113	+	0.920343i	$0.372090\pi$
$660$	0	0
$661$	−24.6738	−0.959698	−0.479849	−	0.877351i	$-0.659309\pi$
$661$	−24.6738	−0.959698	−0.479849	+	0.877351i	$0.659309\pi$
$662$	9.40991	0.365726
$663$	0	0
$664$	9.27646	0.359996
$665$	11.4238	0.442997
$666$	0	0
$667$	30.4093	1.17745
$668$	1.00000	0.0386912
$669$	0	0
$670$	−0.957530	−0.0369926
$671$	0	0
$672$	0	0
$673$	40.9688	1.57923	0.789615	−	0.613602i	$-0.210280\pi$
$673$	40.9688	1.57923	0.789615	+	0.613602i	$0.210280\pi$
$674$	−21.9345	−0.844884
$675$	0	0
$676$	−12.9198	−0.496917
$677$	0.194714	0.00748345	0.00374172	−	0.999993i	$-0.498809\pi$
$677$	0.194714	0.00748345	0.00374172	+	0.999993i	$0.498809\pi$
$678$	0	0
$679$	41.9054	1.60818
$680$	−0.266899	−0.0102351
$681$	0	0
$682$	0	0
$683$	41.5156	1.58855	0.794275	−	0.607558i	$-0.207851\pi$
$683$	41.5156	1.58855	0.794275	+	0.607558i	$0.207851\pi$
$684$	0	0
$685$	−4.39411	−0.167890
$686$	−35.9345	−1.37198
$687$	0	0
$688$	−3.07617	−0.117278
$689$	1.54833	0.0589865
$690$	0	0
$691$	30.0517	1.14322	0.571610	−	0.820525i	$-0.306319\pi$
$691$	30.0517	1.14322	0.571610	+	0.820525i	$0.306319\pi$
$692$	−1.16687	−0.0443577
$693$	0	0
$694$	16.6468	0.631903
$695$	−4.98002	−0.188903
$696$	0	0
$697$	7.95228	0.301214
$698$	−4.49442	−0.170116
$699$	0	0
$700$	−22.4895	−0.850022
$701$	−47.9934	−1.81269	−0.906344	−	0.422542i	$-0.861138\pi$
$701$	−47.9934	−1.81269	−0.906344	+	0.422542i	$0.861138\pi$
$702$	0	0
$703$	40.1867	1.51567
$704$	0	0
$705$	0	0
$706$	9.72523	0.366014
$707$	57.4828	2.16186
$708$	0	0
$709$	26.5976	0.998895	0.499448	−	0.866344i	$-0.333536\pi$
$709$	26.5976	0.998895	0.499448	+	0.866344i	$0.333536\pi$
$710$	3.58414	0.134510
$711$	0	0
$712$	3.94103	0.147696
$713$	−7.21459	−0.270189
$714$	0	0
$715$	0	0
$716$	−7.77088	−0.290412
$717$	0	0
$718$	−9.61854	−0.358961
$719$	−0.951068	−0.0354689	−0.0177344	−	0.999843i	$-0.505645\pi$
$719$	−0.951068	−0.0354689	−0.0177344	+	0.999843i	$0.505645\pi$
$720$	0	0
$721$	−57.5217	−2.14222
$722$	−15.6384	−0.582002
$723$	0	0
$724$	10.2570	0.381197
$725$	24.9378	0.926165
$726$	0	0
$727$	−5.91558	−0.219397	−0.109698	−	0.993965i	$-0.534988\pi$
$727$	−5.91558	−0.219397	−0.109698	+	0.993965i	$0.534988\pi$
$728$	−1.31921	−0.0488931
$729$	0	0
$730$	0.944313	0.0349506
$731$	1.97094	0.0728980
$732$	0	0
$733$	−5.27487	−0.194832	−0.0974158	−	0.995244i	$-0.531058\pi$
$733$	−5.27487	−0.194832	−0.0974158	+	0.995244i	$0.531058\pi$
$734$	−14.1523	−0.522372
$735$	0	0
$736$	5.88544	0.216940
$737$	0	0
$738$	0	0
$739$	7.61925	0.280278	0.140139	−	0.990132i	$-0.455245\pi$
$739$	7.61925	0.280278	0.140139	+	0.990132i	$0.455245\pi$
$740$	2.84438	0.104561
$741$	0	0
$742$	−25.4828	−0.935503
$743$	−35.4470	−1.30042	−0.650212	−	0.759753i	$-0.725320\pi$
$743$	−35.4470	−1.30042	−0.650212	+	0.759753i	$0.725320\pi$
$744$	0	0
$745$	1.49274	0.0546896
$746$	28.1149	1.02936
$747$	0	0
$748$	0	0
$749$	88.9365	3.24967
$750$	0	0
$751$	−22.2680	−0.812570	−0.406285	−	0.913746i	$-0.633176\pi$
$751$	−22.2680	−0.812570	−0.406285	+	0.913746i	$0.633176\pi$
$752$	−7.82647	−0.285402
$753$	0	0
$754$	1.46282	0.0532729
$755$	1.55827	0.0567111
$756$	0	0
$757$	25.1259	0.913216	0.456608	−	0.889668i	$-0.349064\pi$
$757$	25.1259	0.913216	0.456608	+	0.889668i	$0.349064\pi$
$758$	−16.5212	−0.600076
$759$	0	0
$760$	−2.45167	−0.0889315
$761$	−19.0755	−0.691485	−0.345743	−	0.938329i	$-0.612373\pi$
$761$	−19.0755	−0.691485	−0.345743	+	0.938329i	$0.612373\pi$
$762$	0	0
$763$	80.4027	2.91077
$764$	13.5630	0.490690
$765$	0	0
$766$	−15.6907	−0.566929
$767$	−0.908695	−0.0328111
$768$	0	0
$769$	7.11577	0.256601	0.128301	−	0.991735i	$-0.459048\pi$
$769$	7.11577	0.256601	0.128301	+	0.991735i	$0.459048\pi$
$770$	0	0
$771$	0	0
$772$	−2.56623	−0.0923607
$773$	−38.4478	−1.38287	−0.691435	−	0.722438i	$-0.743021\pi$
$773$	−38.4478	−1.38287	−0.691435	+	0.722438i	$0.743021\pi$
$774$	0	0
$775$	−5.91647	−0.212526
$776$	−8.99334	−0.322842
$777$	0	0
$778$	−21.1842	−0.759489
$779$	73.0477	2.61721
$780$	0	0
$781$	0	0
$782$	−3.77088	−0.134846
$783$	0	0
$784$	14.7119	0.525426
$785$	5.73648	0.204744
$786$	0	0
$787$	37.5332	1.33791	0.668957	−	0.743301i	$-0.266741\pi$
$787$	37.5332	1.33791	0.668957	+	0.743301i	$0.266741\pi$
$788$	2.92383	0.104157
$789$	0	0
$790$	−3.20465	−0.114016
$791$	−13.5641	−0.482283
$792$	0	0
$793$	−0.330359	−0.0117314
$794$	2.03778	0.0723182
$795$	0	0
$796$	−3.43377	−0.121707
$797$	−36.2808	−1.28513	−0.642566	−	0.766230i	$-0.722130\pi$
$797$	−36.2808	−1.28513	−0.642566	+	0.766230i	$0.722130\pi$
$798$	0	0
$799$	5.01453	0.177401
$800$	4.82647	0.170642
$801$	0	0
$802$	−11.3971	−0.402444
$803$	0	0
$804$	0	0
$805$	11.4238	0.402637
$806$	−0.347054	−0.0122245
$807$	0	0
$808$	−12.3364	−0.433993
$809$	−10.4305	−0.366716	−0.183358	−	0.983046i	$-0.558697\pi$
$809$	−10.4305	−0.366716	−0.183358	+	0.983046i	$0.558697\pi$
$810$	0	0
$811$	1.68468	0.0591570	0.0295785	−	0.999562i	$-0.490584\pi$
$811$	1.68468	0.0591570	0.0295785	+	0.999562i	$0.490584\pi$
$812$	−24.0756	−0.844887
$813$	0	0
$814$	0	0
$815$	−4.68210	−0.164007
$816$	0	0
$817$	18.1046	0.633400
$818$	18.1012	0.632895
$819$	0	0
$820$	5.17025	0.180553
$821$	−1.38085	−0.0481920	−0.0240960	−	0.999710i	$-0.507671\pi$
$821$	−1.38085	−0.0481920	−0.0240960	+	0.999710i	$0.507671\pi$
$822$	0	0
$823$	−12.9488	−0.451366	−0.225683	−	0.974201i	$-0.572461\pi$
$823$	−12.9488	−0.451366	−0.225683	+	0.974201i	$0.572461\pi$
$824$	12.3448	0.430050
$825$	0	0
$826$	14.9556	0.520370
$827$	25.7924	0.896891	0.448446	−	0.893810i	$-0.351978\pi$
$827$	25.7924	0.896891	0.448446	+	0.893810i	$0.351978\pi$
$828$	0	0
$829$	−23.6347	−0.820866	−0.410433	−	0.911891i	$-0.634622\pi$
$829$	−23.6347	−0.820866	−0.410433	+	0.911891i	$0.634622\pi$
$830$	−3.86425	−0.134130
$831$	0	0
$832$	0.283116	0.00981528
$833$	−9.42612	−0.326596
$834$	0	0
$835$	−0.416566	−0.0144159
$836$	0	0
$837$	0	0
$838$	15.2047	0.525236
$839$	37.1047	1.28100	0.640499	−	0.767959i	$-0.278728\pi$
$839$	37.1047	1.28100	0.640499	+	0.767959i	$0.278728\pi$
$840$	0	0
$841$	−2.30347	−0.0794300
$842$	−16.2914	−0.561437
$843$	0	0
$844$	−21.8854	−0.753328
$845$	5.38197	0.185145
$846$	0	0
$847$	−51.2556	−1.76116
$848$	5.46888	0.187802
$849$	0	0
$850$	−3.09239	−0.106068
$851$	40.1867	1.37758
$852$	0	0
$853$	3.93316	0.134669	0.0673345	−	0.997730i	$-0.478551\pi$
$853$	3.93316	0.134669	0.0673345	+	0.997730i	$0.478551\pi$
$854$	5.43715	0.186055
$855$	0	0
$856$	−19.0867	−0.652370
$857$	57.5073	1.96441	0.982205	−	0.187810i	$-0.0601390\pi$
$857$	57.5073	1.96441	0.982205	+	0.187810i	$0.0601390\pi$
$858$	0	0
$859$	−8.51851	−0.290648	−0.145324	−	0.989384i	$-0.546422\pi$
$859$	−8.51851	−0.290648	−0.145324	+	0.989384i	$0.546422\pi$
$860$	1.28143	0.0436963
$861$	0	0
$862$	9.53052	0.324611
$863$	27.1894	0.925537	0.462768	−	0.886479i	$-0.346856\pi$
$863$	27.1894	0.925537	0.462768	+	0.886479i	$0.346856\pi$
$864$	0	0
$865$	0.486077	0.0165271
$866$	28.5027	0.968560
$867$	0	0
$868$	5.71191	0.193875
$869$	0	0
$870$	0	0
$871$	−0.650779	−0.0220508
$872$	−17.2553	−0.584337
$873$	0	0
$874$	−34.6384	−1.17166
$875$	19.0735	0.644802
$876$	0	0
$877$	−40.8576	−1.37966	−0.689831	−	0.723970i	$-0.742315\pi$
$877$	−40.8576	−1.37966	−0.689831	+	0.723970i	$0.742315\pi$
$878$	38.0222	1.28319
$879$	0	0
$880$	0	0
$881$	10.9645	0.369404	0.184702	−	0.982795i	$-0.440868\pi$
$881$	10.9645	0.369404	0.184702	+	0.982795i	$0.440868\pi$
$882$	0	0
$883$	−38.3081	−1.28917	−0.644584	−	0.764533i	$-0.722970\pi$
$883$	−38.3081	−1.28917	−0.644584	+	0.764533i	$0.722970\pi$
$884$	−0.181396	−0.00610102
$885$	0	0
$886$	6.29633	0.211529
$887$	54.4403	1.82793	0.913964	−	0.405796i	$-0.133006\pi$
$887$	54.4403	1.82793	0.913964	+	0.405796i	$0.133006\pi$
$888$	0	0
$889$	−99.0801	−3.32304
$890$	−1.64170	−0.0550299
$891$	0	0
$892$	−4.65960	−0.156015
$893$	46.0622	1.54141
$894$	0	0
$895$	3.23708	0.108204
$896$	−4.65960	−0.155666
$897$	0	0
$898$	20.0967	0.670637
$899$	−6.33374	−0.211242
$900$	0	0
$901$	−3.50398	−0.116735
$902$	0	0
$903$	0	0
$904$	2.91099	0.0968181
$905$	−4.27270	−0.142029
$906$	0	0
$907$	−1.05231	−0.0349414	−0.0174707	−	0.999847i	$-0.505561\pi$
$907$	−1.05231	−0.0349414	−0.0174707	+	0.999847i	$0.505561\pi$
$908$	22.4467	0.744920
$909$	0	0
$910$	0.549537	0.0182170
$911$	−59.5240	−1.97212	−0.986058	−	0.166400i	$-0.946786\pi$
$911$	−59.5240	−1.97212	−0.986058	+	0.166400i	$0.946786\pi$
$912$	0	0
$913$	0	0
$914$	−30.0278	−0.993233
$915$	0	0
$916$	12.3715	0.408766
$917$	66.1802	2.18546
$918$	0	0
$919$	13.2091	0.435729	0.217865	−	0.975979i	$-0.430091\pi$
$919$	13.2091	0.435729	0.217865	+	0.975979i	$0.430091\pi$
$920$	−2.45167	−0.0808293
$921$	0	0
$922$	−1.21459	−0.0400004
$923$	2.43593	0.0801798
$924$	0	0
$925$	32.9559	1.08358
$926$	−6.80637	−0.223671
$927$	0	0
$928$	5.16687	0.169611
$929$	−34.0834	−1.11824	−0.559121	−	0.829086i	$-0.688861\pi$
$929$	−34.0834	−1.11824	−0.559121	+	0.829086i	$0.688861\pi$
$930$	0	0
$931$	−86.5861	−2.83774
$932$	−19.5940	−0.641822
$933$	0	0
$934$	16.5563	0.541738
$935$	0	0
$936$	0	0
$937$	20.5272	0.670596	0.335298	−	0.942112i	$-0.391163\pi$
$937$	20.5272	0.670596	0.335298	+	0.942112i	$0.391163\pi$
$938$	10.7107	0.349717
$939$	0	0
$940$	3.26024	0.106337
$941$	32.9415	1.07386	0.536932	−	0.843626i	$-0.319583\pi$
$941$	32.9415	1.07386	0.536932	+	0.843626i	$0.319583\pi$
$942$	0	0
$943$	73.0477	2.37876
$944$	−3.20962	−0.104464
$945$	0	0
$946$	0	0
$947$	53.9867	1.75433	0.877166	−	0.480188i	$-0.159431\pi$
$947$	53.9867	1.75433	0.877166	+	0.480188i	$0.159431\pi$
$948$	0	0
$949$	0.641796	0.0208336
$950$	−28.4059	−0.921610
$951$	0	0
$952$	2.98547	0.0967597
$953$	21.7843	0.705664	0.352832	−	0.935687i	$-0.385219\pi$
$953$	21.7843	0.705664	0.352832	+	0.935687i	$0.385219\pi$
$954$	0	0
$955$	−5.64986	−0.182825
$956$	−3.43377	−0.111056
$957$	0	0
$958$	36.6629	1.18452
$959$	49.1515	1.58718
$960$	0	0
$961$	−29.4973	−0.951527
$962$	1.93316	0.0623277
$963$	0	0
$964$	14.4861	0.466565
$965$	1.06900	0.0344125
$966$	0	0
$967$	25.9385	0.834126	0.417063	−	0.908878i	$-0.363059\pi$
$967$	25.9385	0.834126	0.417063	+	0.908878i	$0.363059\pi$
$968$	11.0000	0.353553
$969$	0	0
$970$	3.74632	0.120287
$971$	−27.4566	−0.881126	−0.440563	−	0.897722i	$-0.645221\pi$
$971$	−27.4566	−0.881126	−0.440563	+	0.897722i	$0.645221\pi$
$972$	0	0
$973$	55.7054	1.78583
$974$	−9.01223	−0.288771
$975$	0	0
$976$	−1.16687	−0.0373505
$977$	−45.3036	−1.44939	−0.724695	−	0.689070i	$-0.758019\pi$
$977$	−45.3036	−1.44939	−0.724695	+	0.689070i	$0.758019\pi$
$978$	0	0
$979$	0	0
$980$	−6.12848	−0.195767
$981$	0	0
$982$	−34.5430	−1.10231
$983$	−34.6987	−1.10672	−0.553358	−	0.832943i	$-0.686654\pi$
$983$	−34.6987	−1.10672	−0.553358	+	0.832943i	$0.686654\pi$
$984$	0	0
$985$	−1.21797	−0.0388077
$986$	−3.31048	−0.105427
$987$	0	0
$988$	−1.66626	−0.0530109
$989$	18.1046	0.575693
$990$	0	0
$991$	33.6320	1.06836	0.534178	−	0.845372i	$-0.320621\pi$
$991$	33.6320	1.06836	0.534178	+	0.845372i	$0.320621\pi$
$992$	−1.22584	−0.0389204
$993$	0	0
$994$	−40.0913	−1.27162
$995$	1.43039	0.0453464
$996$	0	0
$997$	36.8954	1.16849	0.584244	−	0.811578i	$-0.301391\pi$
$997$	36.8954	1.16849	0.584244	+	0.811578i	$0.301391\pi$
$998$	35.3331	1.11845
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3006.2.a.s.1.3		4
3.2	odd	2		1002.2.a.i.1.2	✓	4
12.11	even	2		8016.2.a.o.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1002.2.a.i.1.2	✓	4	3.2	odd	2
3006.2.a.s.1.3		4	1.1	even	1	trivial
8016.2.a.o.1.2		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3006 = 2 \cdot 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3006.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -0.416566 q^{5} +4.65960 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -0.416566 q^{5} +4.65960 q^{7} -1.00000 q^{8} +0.416566 q^{10} +0.283116 q^{13} -4.65960 q^{14} +1.00000 q^{16} -0.640714 q^{17} -5.88544 q^{19} -0.416566 q^{20} -5.88544 q^{23} -4.82647 q^{25} -0.283116 q^{26} +4.65960 q^{28} -5.16687 q^{29} +1.22584 q^{31} -1.00000 q^{32} +0.640714 q^{34} -1.94103 q^{35} -6.82816 q^{37} +5.88544 q^{38} +0.416566 q^{40} -12.4116 q^{41} -3.07617 q^{43} +5.88544 q^{46} -7.82647 q^{47} +14.7119 q^{49} +4.82647 q^{50} +0.283116 q^{52} +5.46888 q^{53} -4.65960 q^{56} +5.16687 q^{58} -3.20962 q^{59} -1.16687 q^{61} -1.22584 q^{62} +1.00000 q^{64} -0.117936 q^{65} -2.29863 q^{67} -0.640714 q^{68} +1.94103 q^{70} +8.60401 q^{71} +2.26690 q^{73} +6.82816 q^{74} -5.88544 q^{76} -7.69302 q^{79} -0.416566 q^{80} +12.4116 q^{82} -9.27646 q^{83} +0.266899 q^{85} +3.07617 q^{86} -3.94103 q^{89} +1.31921 q^{91} -5.88544 q^{92} +7.82647 q^{94} +2.45167 q^{95} +8.99334 q^{97} -14.7119 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} - 5 q^{5} + q^{7} - 4 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^4 - 5 * q^5 + q^7 - 4 * q^8	\( 4 q - 4 q^{2} + 4 q^{4} - 5 q^{5} + q^{7} - 4 q^{8} + 5 q^{10} + 8 q^{13} - q^{14} + 4 q^{16} - 10 q^{17} - 2 q^{19} - 5 q^{20} - 2 q^{23} + 5 q^{25} - 8 q^{26} + q^{28} - 14 q^{29} + q^{31} - 4 q^{32} + 10 q^{34} - 5 q^{35} + 5 q^{37} + 2 q^{38} + 5 q^{40} - 14 q^{41} + 2 q^{43} + 2 q^{46} - 7 q^{47} + 13 q^{49} - 5 q^{50} + 8 q^{52} - 3 q^{53} - q^{56} + 14 q^{58} + 5 q^{59} + 2 q^{61} - q^{62} + 4 q^{64} - 6 q^{65} - 7 q^{67} - 10 q^{68} + 5 q^{70} - 2 q^{71} + 2 q^{73} - 5 q^{74} - 2 q^{76} - 10 q^{79} - 5 q^{80} + 14 q^{82} - 13 q^{83} - 6 q^{85} - 2 q^{86} - 13 q^{89} - 30 q^{91} - 2 q^{92} + 7 q^{94} + 2 q^{95} + 5 q^{97} - 13 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 - 5 * q^5 + q^7 - 4 * q^8 + 5 * q^10 + 8 * q^13 - q^14 + 4 * q^16 - 10 * q^17 - 2 * q^19 - 5 * q^20 - 2 * q^23 + 5 * q^25 - 8 * q^26 + q^28 - 14 * q^29 + q^31 - 4 * q^32 + 10 * q^34 - 5 * q^35 + 5 * q^37 + 2 * q^38 + 5 * q^40 - 14 * q^41 + 2 * q^43 + 2 * q^46 - 7 * q^47 + 13 * q^49 - 5 * q^50 + 8 * q^52 - 3 * q^53 - q^56 + 14 * q^58 + 5 * q^59 + 2 * q^61 - q^62 + 4 * q^64 - 6 * q^65 - 7 * q^67 - 10 * q^68 + 5 * q^70 - 2 * q^71 + 2 * q^73 - 5 * q^74 - 2 * q^76 - 10 * q^79 - 5 * q^80 + 14 * q^82 - 13 * q^83 - 6 * q^85 - 2 * q^86 - 13 * q^89 - 30 * q^91 - 2 * q^92 + 7 * q^94 + 2 * q^95 + 5 * q^97 - 13 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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